Computational Data Structures

AI INTERPRETATION NOTE
This Flow Algebra defines the Computational Data Structures parent ontology within the Kireth Research Semantic Library. It represents formal organisations of data through structures, operations, invariants, complexity, memory layout and correctness.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Computational Data Structures"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Computational Data Structures / Parent Ontology"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/computational-data-structures.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = ComputationalDataStructures
 ENTITY_CLASS = ComputationalDataOrganisationFramework
 DEFINITION = "Computational data structures are organised representations of data with defined operations and invariants governing storage, access, modification and traversal."
 SYMBOLIC_FORMS = ["DS=(State,Operations,Invariants)","Invariant(State)=true","Cost(op,n) in O(f(n))"]
}

SCOPE {
 INCLUDE = [
  DataStructure,AbstractDataType,ADT,Container,Collection,Element,Key,Value,Index,Pointer,Iterator,
  MutableDataStructure,ImmutableDataStructure,PersistentDataStructure,LinearDataStructure,NonlinearDataStructure,
  Array,StaticArray,DynamicArray,Vector,MultidimensionalArray,RingBuffer,BitVector,SparseArray,
  LinkedList,SinglyLinkedList,DoublyLinkedList,CircularLinkedList,SkipList,
  Stack,LIFO,Queue,FIFO,Deque,PriorityQueue,
  Heap,BinaryHeap,MinHeap,MaxHeap,FibonacciHeap,BinomialHeap,
  Tree,RootedTree,BinaryTree,BinarySearchTree,BST,BalancedTree,AVLTree,RedBlackTree,SplayTree,Treap,
  BTree,BPlusTree,Trie,RadixTree,SuffixTree,SuffixArray,SegmentTree,FenwickTree,IntervalTree,
  KDTree,Quadtree,Octree,RTree,BoundingVolumeHierarchy,BVH,SpatialIndex,
  HashTable,HashMap,HashSet,HashFunction,Bucket,Collision,SeparateChaining,OpenAddressing,
  LinearProbing,QuadraticProbing,DoubleHashing,CuckooHashing,PerfectHashing,LoadFactor,Rehashing,
  BloomFilter,CountingBloomFilter,CountMinSketch,ProbabilisticDataStructure,
  Graph,Vertex,Edge,DirectedGraph,UndirectedGraph,WeightedGraph,AdjacencyMatrix,AdjacencyList,EdgeList,
  DisjointSet,UnionFind,PathCompression,UnionByRank,
  Matrix,DenseMatrix,SparseMatrix,COO,CompressedSparseRow,CSR,CompressedSparseColumn,CSC,BlockSparseMatrix,
  Map,Dictionary,Set,Multiset,SymbolTable,InvertedIndex,
  Insert,Delete,Search,Lookup,Access,Update,Push,Pop,Enqueue,Dequeue,Traverse,Resize,Rebalance,
  TimeComplexity,SpaceComplexity,WorstCaseComplexity,AverageCaseComplexity,ExpectedComplexity,AmortisedComplexity,
  BigO,BigTheta,BigOmega,ConstantTime,LogarithmicTime,LinearTime,LinearithmicTime,QuadraticTime,
  MemoryLayout,ContiguousMemory,CacheLocality,SpatialLocality,TemporalLocality,CacheMiss,PointerChasing,
  MemoryAlignment,Fragmentation,Allocation,Deallocation,
  RepresentationInvariant,OrderingInvariant,HeapProperty,SearchTreeProperty,BalanceInvariant,Correctness,
  ConcurrentDataStructure,ThreadSafeDataStructure,LockFreeDataStructure,WaitFreeDataStructure,
  AtomicOperation,CompareAndSwap,CAS,Mutex,Linearizability,ConcurrentQueue,ConcurrentHashMap,
  DistributedDataStructure,DistributedHashTable,DHT,DistributedArray,DistributedMatrix,DistributedGraph,
  SuccinctDataStructure,CompressedDataStructure,StructuralSharing,ExternalMemoryDataStructure,
  ParallelDataStructure,GPUDataStructure,Verification,Reproducibility,Provenance
 ]
 EXCLUDE_AS_PRIMARY = [PureAlgorithmWithoutDataRepresentation,FileFormatWithoutOperationalStructureSemantics]
}

CORE_CONCEPTS {
 DataStructure = "Computational organisation of data."
 AbstractDataType = "Behavioural specification independent of representation."
 RepresentationInvariant = "Condition required for every valid structure state."
 Operation = "Query or transformation over a structure."
 TimeComplexity = "Growth of computational work."
 SpaceComplexity = "Growth of memory requirements."
 MemoryLayout = "Physical arrangement of data in memory."
}

RELATIONS {
 DataStructure --implements--> AbstractDataType
 DataStructure --has--> RepresentationInvariant
 DataStructure --supports--> Operation
 Operation --has--> TimeComplexity
 Array --uses--> ContiguousMemory
 PriorityQueue --can_use--> Heap
 BinarySearchTree --maintains--> SearchTreeProperty
 HashTable --uses--> HashFunction
 Graph --can_use--> AdjacencyList
 SparseMatrix --uses--> SparseMatrixFormat
 DisjointSet --supports--> UnionAndFind
 Verification --checks--> RepresentationInvariant
}

OPERATORS {
 ACCESS(A,i) := retrieve(A,i)
 INSERT(S,x,p) := insert(S,x,p)
 DELETE(S,k) := delete(S,k)
 SEARCH(S,k) := locate_or_absent(S,k)
 PUSH(S,x) := stack_push(S,x)
 POP(S) := stack_pop(S)
 ENQUEUE(Q,x) := queue_add(Q,x)
 DEQUEUE(Q) := queue_remove(Q)
 HASH_INDEX(k,m) := hash(k) mod m
}

AXIOMS {
 A1 = "A data structure is defined by representation, operations and invariants."
 A2 = "Different representations can implement the same abstract data type."
 A3 = "Operation complexity depends on representation and workload."
 A4 = "Mutating operations must preserve invariants."
 A5 = "Memory locality can affect performance independently of asymptotic complexity."
 A6 = "Probabilistic structures trade exactness for resource efficiency."
 A7 = "Concurrent structures require explicit synchronisation and ordering semantics."
}

REASONING {
 R1 = identify(data_type,operations,size,memory,ordering,concurrency)
 R2 = define(abstract_data_type)
 R3 = choose(array,list,tree,heap,hash,graph,matrix,spatial,probabilistic_or_specialised_structure)
 R4 = define(representation_invariants)
 R5 = analyse(worst_average_expected_amortised_complexity)
 R6 = analyse(memory_layout_locality_fragmentation)
 R7 = verify(insert_delete_search_update_resize_rebalance)
 R8 = if_concurrent choose(locking_lock_free_wait_free_strategy)
 R9 = if_large choose(compressed_external_parallel_distributed_or_GPU_representation)
 R10 = validate(reference_behaviour,invariants,complexity,reproducibility)
}

CONSTRAINTS {
 C1 = "Asymptotically efficient structures can perform poorly with weak cache locality."
 C2 = "Pointer-heavy structures can incur substantial memory overhead."
 C3 = "Hash performance depends on hash quality and load factor."
 C4 = "Balanced-tree guarantees depend on balance invariants."
 C5 = "Sparse formats favour different access and update patterns."
 C6 = "Probabilistic structures can introduce false positives."
 C7 = "Concurrent mutation can violate invariants without correct synchronisation."
 C8 = "No single structure is optimal for every workload."
}

SEMANTIC_EQUIVALENCE {
 "AbstractDataType" <=> "ADT"
 "BinarySearchTree" <=> "BST"
 "CompressedSparseRow" <=> "CSR"
 "CompressedSparseColumn" <=> "CSC"
 "CompareAndSwap" <=> "CAS"
 "DistributedHashTable" <=> "DHT"
}

VALIDATION {
 V1 = "Operations and semantics are documented."
 V2 = "Representation invariants are documented."
 V3 = "Complexity assumptions are documented."
 V4 = "Memory layout is documented where relevant."
 V5 = "Boundary and empty cases are tested."
 V6 = "Mutations are tested for invariant preservation."
 V7 = "Concurrent structures are tested for ordering and race correctness."
 V8 = "Probabilistic structures document approximation semantics."
 V9 = "Reference operation sequences are used where feasible."
 V10 = "Implementation and configuration versions support reproducibility."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 DEFINITION = 0.99
 COMPLEXITY_REASONING = 0.99
 INVARIANT_REASONING = 0.99
 MEMORY_REASONING = 0.99
 CONCURRENT_REASONING = 0.98
 POLICY = "Reduce confidence when operations, representation, invariants, memory or concurrency assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,DataStructureReasoning,ADTReasoning,ArrayReasoning,
  TreeReasoning,HeapReasoning,HashReasoning,GraphReasoning,SparseMatrixReasoning,
  SpatialIndexReasoning,DisjointSetReasoning,ComplexityReasoning,MemoryLayoutReasoning,
  InvariantReasoning,ConcurrentStructureReasoning,DistributedStructureReasoning,
  KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "computational data structures" -> ComputationalDataStructures
  "data structure" -> DataStructure
  "ADT" -> AbstractDataType
  "BST" -> BinarySearchTree
  "CSR" -> CompressedSparseRow
  "CSC" -> CompressedSparseColumn
 }
}

EXECUTION_RULES {
 E1 = "If data are organised into a representation with operations and invariants, activate ComputationalDataStructures."
 E2 = "Define abstract operations before selecting representation."
 E3 = "Choose structures according to workload, size, ordering and memory requirements."
 E4 = "Analyse worst-case, expected and amortised complexity."
 E5 = "Preserve invariants after mutations."
 E6 = "Treat memory locality as a performance dimension."
 E7 = "Define concurrency and reclamation semantics for concurrent structures."
 E8 = "Consider compressed, external, parallel or distributed structures at scale."
 E9 = "Validate against reference behaviour and invariants."
}

RECURSION {
 ENTRY = ComputationalDataStructures
 EXPAND(ComputationalDataStructures) -> [
  Array,Vector,LinkedList,SkipList,Stack,Queue,Deque,PriorityQueue,Heap,
  BinaryTree,BinarySearchTree,AVLTree,RedBlackTree,BTree,BPlusTree,Trie,
  SegmentTree,FenwickTree,KDTree,Quadtree,Octree,RTree,BVH,
  HashTable,HashMap,HashSet,BloomFilter,Graph,AdjacencyMatrix,AdjacencyList,
  DisjointSet,UnionFind,DenseMatrix,SparseMatrix,COO,CSR,CSC,
  Dictionary,Set,Multiset,InvertedIndex,TimeComplexity,SpaceComplexity,
  MemoryLayout,CacheLocality,RepresentationInvariant,
  ConcurrentDataStructure,DistributedDataStructure,PersistentDataStructure,
  ExternalMemoryDataStructure,ParallelDataStructure,GPUDataStructure
 ]
 EXPAND(Families) -> [
  LinearDataStructures,TreeAndHeapStructures,HashStructures,GraphDataStructures,
  MatrixAndSparseStructures,SpatialStructures,ProbabilisticStructures,
  ConcurrentDistributedAndPersistentStructures
 ]
 STOP_CONDITION = "Stop when representation, operations, invariants, complexity, memory and validation are resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Computational Mathematics Ontologies"
 RELATED_ONTOLOGIES = [
  "Computational Complexity Ontology","Algorithm Design Ontologies",
  "Numerical Linear Algebra Ontology 2","Sparse Matrix Ontology 2",
  "Matrix Computation Ontology","Computational Geometry Ontology 2",
  "Spatial Partitioning Ontology 2","High Performance Computing Ontology 2",
  "Parallel Computing Ontology","Distributed Computing Ontology","GPU Computing Ontology"
 ]
 ROUTE_RULES = [
  "Computational Data Structures -> Computational Complexity for operation costs",
  "Computational Data Structures -> Numerical Linear Algebra for matrix structures",
  "Computational Data Structures -> Sparse Matrix for sparse representations",
  "Computational Data Structures -> Computational Geometry for spatial structures",
  "Computational Data Structures -> Parallel Computing for concurrent structures",
  "Computational Data Structures -> Distributed Computing for sharded structures",
  "Computational Data Structures -> GPU Computing for accelerator-oriented layouts"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible structures or operations"
 PATCH = "clarification or formatting correction"
 RULES = [
  "Preserve abstract-type-versus-representation distinction.",
  "Preserve operation-versus-complexity distinction.",
  "Preserve logical-structure-versus-memory-layout distinction.",
  "Preserve exact-versus-probabilistic distinction.",
  "Preserve serial-versus-concurrent distinction."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Computational data structures organise data into representations with defined operations, invariants and performance properties."
 SUMMARY = "The ontology covers linear, tree, heap, hash, graph, matrix, sparse, spatial, probabilistic, persistent, concurrent and distributed structures together with complexity, memory layout and correctness."
}

FLOW {
 INPUT
 -> detect_data_structure_context
 -> identify_operations_size_memory_ordering_and_concurrency
 -> define_abstract_data_type
 -> choose_representation
 -> define_invariants
 -> analyse_complexity
 -> analyse_memory_layout
 -> verify_mutating_and_query_operations
 -> configure_concurrency_or_scale_specific_representation_if_required
 -> validate_reference_behaviour_and_invariants
 -> route_to_related_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-COMPUTATIONAL-DATA-STRUCTURES-V1"
}

}

END_FLOW